Ning Wu Title and Abstract

Capacity of Shared-Short Lanes at Unsignalised Intersections

by Ning Wu

(Publisch in Proceedings of the Third International Symposium on Intersections Without

Traffic Signals. Portland, Oregen, July 1997. University of Idaho, Moscow, 1997)

Author's address: Dr. Ning Wu

Institute for Transportation

Ruhr-University Bochum

44780 Bochum, Germany

Tel.: ++49/234/7006557

Fax: ++49/234/7094151

E-mail: Ning.Wu@rz.ruhr-uni-bochum.de

Ning Wu Title and Abstract

ABSTRACT

The calculation procedures in recent highway capacity manuals do not exactly treat shared - short

lanes at intersections without traffic signals. The capacity of individual streams (left turn, straight

ahead and right turn) are calculated separately. If the streams share a common traffic lane, the

capacity of the shared lane is then calculated according to the shared lane procedure from Harders

(1968). That means, the lengths of the short lanes are considered either as infinite or as zero. The

exact lengths of the separate short lanes cannot be take in account. Therefore, the capacity computed

from conventional methods is overestimated, whereas that from the shared lanes formula - like in

chapter 10 of the HCM, 1994, is underestimated.

This paper presents an analytical theory for estimating the capacity of this combination of shared and

short lanes. It is based on probability theory. This theory combines the existing procedures for

estimating the capacity of shared and short lanes. It was checked by simulations in the style of the

KNOSIMO-simulation. This theory can be used for arbitrary lane configurations. For the simple

shared-short lane configurations, explicit equations are derived for estimating the capacity. For

complicated shared-short lane configurations, iteration procedures are given. For practical

applications a graph, which should facilitate the iterations needed by calculating capacity of

complicated shared-short lanes, is prepared.

As a special case, the so-called flared minor approaches are treated according to the theory derived.

keywords: capacity, unsignalised intersection, short lanes, shared lanes, flared lanes.

Ning Wu 1

1. INTRODUCTION

Intersections (cross-roads and T-junctions) without traffic signals are the mostly used intersections in

traffic management. The traffic controls at these intersections are regulated by traffic signs.

The right of way regulated by traffic signs presupposes that a driver makes the decision for passing

through if he is at the first waiting position directly at the stop line or if in front of him no other

vehicle is waiting. The calculation procedures developed for this situation, which are also used in

numerous manuals1,2,3), are standard for calculating the capacity of unsignalised intersections. The two

most known and simplest procedures are these from Harders4) and Siegloch5).

The calculation procedures in recent manuals1,2,3) assume that, firstly, the traffic streams, which have

to give way, possess their own traffic lanes at the intersection. The capacity of the individual streams

(left turn, straight ahead and right turn) are calculated separately. If the streams share a common

traffic lane, the capacity of the shared lane is then calculated according to the shared lane procedure

from Harders4).

The procedures for considering the lane distribution at intersections without traffic signals are: for the

left turn and/or right turn streams either there are infinitely long exclusive lanes or there are no

exclusive lanes at all. However, the length of the turn lanes in reality cannot be considered, i.e., if a

approach with short traffic lanes (cf. Fig.1a) for the left turn and/or right turn streams is calculated, the

capacity is either overestimated (length of the exclusive lanes as infinite) or underestimated (length of

the exclusive lanes as zero).

In this paper a procedure is derived, with which the length of the turn lanes can be considered exactly

for calculating the capacity of the shared lane. The precision of this calculation procedure is examined

by simulations.

Ning Wu 2

In this paper the following symbols and indices are used:

Symbols:

m = number of sub - streams [-]

L = capacity [veh/h]

q = traffic flow [veh/h]

x = saturation degree [-]

n = length of queue space in number of vehicles [veh]

Ps = probability that a point on the street is occupied by traffic [-]

k = factor for estimating the capacity of shared lane = 1 / xsh,real [-]

xsh,real = real saturation degree of shared lane = qsh / Lsh [-]

Lsh = capacity of shared lane [veh/h]

qsh = traffic flow of shared lane [veh/h]

xsh = apparent saturation degree of shared lane [-]

Indices for systems with arbitrarily many sub-streams:

i = index for the i-th sub-stream

i1 = index for the i-th sub-stream of the level 1

i2 = index for the i-th sub-stream of the level 2

j = index for the j-th step of iterations

sh = index for shared lane

sh1 = index for shared lane of the level 1

sh2 = index for shared lane of the level 2

Indices for systems with three sub-streams:

L = index for left turn streams and their traffic lanes

G = index for crossing streams and their traffic lanes

R = index for right turn streams and their traffic lanes

Ning Wu 3

LG = index for shared streams consisting of a left turn and a right turn stream and their

traffic lanes

GR = index for shared streams consisting of a crossing and a right turn stream and their

traffic lanes

L,H = index for left turn streams and their traffic lanes on the major street

G,H = index for through streams and their traffic lanes on the major street

R,H = index for right turn streams and their traffic lanes on the major street

LG,H = index for shared streams consisting of a left turn and a through stream and their

traffic lanes on the major street

GR,H = index for shared streams consisting of a through and a right turn stream and their

traffic lanes on the major street

Indices for systems with two sub-streams:

I = index for stream I

II = index for stream II

2. MATHEMATICAL DERIVATIONS

In Fig.1a the possible combinations of short traffic lanes are presented. The short traffic lanes at

intersections without traffic signals have usually two basic forms:

A) All three direction streams divide at a point (cf. Fig.1b, type 1and 4)

B) The streams divide one after another at two points (cf. Fig.1b, type 2 and 3)

For both basic forms of short traffic lanes mathematical derivations are given in this paper.

Ning Wu 4

Fig.1a - Possible queues at the approaches of unsignalised intersections

L G R L G R L G R

Type 1 Type 3Type 2 Type 4

L,HG,HR,H

Fig.1b - Combination forms of short traffic lanes

Firstly, we consider a generalized system with m sub-streams, which all develop at the point A from

one shared lane (cf. Fig.2). The sub-stream i is described by the parameters qi (traffic flow), Li

(capacity) and xi (saturation degree). The capacity Li and the saturation degree xi = qi / Li are

considered under the assumption that there are infinitely many queue places for the subject stream i.

Accordingly, the shared lane has the parameters qsh, Lsh and xsh.

Ning Wu 5

.

.nm

.

.ni

n2

n1

A

qsh, Lsh, xshq1, L1, x1

qi, Li, xi

q2, L2, x2

qm, Lm, xm

Fig.2 - Relationship between a shared lane and its sub - streams

For the point A the following fundamental state condition holds:

The point A is equally occupied from left (shared lane) and from right (all sub-streams)

by waiting vehicles

i.e.: the probability that the point A is occupied on the side of the shared lane, is equal to the

probability that the point A is occupied on the side of the sub-streams. It follows that

P P P P P Ps sh s s s i s m s ii

m

, , , , , ,... ...= + + + + + ==∑1 2

1 (1)

The probability that the point A is occupied by a sub-stream, is equal to the probability that the queue

length in this sub-stream is larger than the length of the queue space (section from the stop line to

point A), i.e., for the sub-stream i,

P N ns i i, Pr( )= > (2)

The distribution function of queue lengths in a waiting stream at intersections without traffic signals

can be represented approximately by the following equation (cf. Wu6)):

F n N n xi i ia bni( ) Pr( ) ( )= ≤ = − +1 1 (3)

Ning Wu 6

with

x qLi

i

i

=

a, b = parameters

Accordingly one obtains

P N n F n xs i i i ia bni

,( )Pr( ) ( )= > = − = +1 1 (4)

Also the M/M/1-queuing system is a good approximation for the queuing system at intersections

without traffic signals (cf. Wu6)). In this case we have a = 1 and b = 1. Thus,

P N n xs i i ini

, Pr( )= > = +1 (5)

For the further derivations the queuing system at intersections without traffic signals is considered as

an M/M/1-queuing system. The resulting deviation can be considered as negligible (cf. Wu6)).

If one considers the point A as a counter in the sense of queuing system, then the probability that the

point A is occupied on the side of the shared lane is equal to the saturation degree of the shared lane,

i.e.,

P N x xs sh sh sh, Pr( )= > = =+0 0 1 (6)

Inserting eq.(6) and eq.(5) in the eq.(1), one obtains

P x xs sh in

i

m

shi

, = =+

=∑ 1

1 (7)

However, here xsh is only the apparent saturation degree of the shared lane. That means,

x qLsh

sh

sh

≠

Ning Wu 7

and accordingly one has also

L qx

q

xsh

sh

sh

ii

m

in

i

mi

≠ = =

+

=

∑

∑1

1

1

The establishment of these inequalities lies therein, that no linear relationship exits between the traffic

flow and the saturation degree in the shared lane due to the exponents of xi. The capacity of the shared

lane can only be determined by other ways.

For estimating the capacity of the shared lane, the following definition is made:

The capacity of the shared lane is the traffic flow, at which the merge point A on both

sides is occupied 100 percent ( P x xs sh sh i, ,max ,max= = =∑ 1).

As a rule, the traffic flows qi (existing or predicted) do not describe the complete saturation of the

shared lane. The capacity of the shared lane lies generally over the sum of qi (in case of under-

saturation by existing qi). In this case the traffic flows at the subject traffic stream would approach the

limit of the capacity, if the qi-values increase. In general, each qi-value could have another increase. It

is assumed however, that for these fictional increases of existing traffic flows equal increase factor k

can be applied. k is thus that factor, by which all traffic flows on the subject approach has to increase,

for reaching just the maximal possible traffic flow: the capacity.

Multiplying the saturation degree of all sub-streams by this factor k and postulating

P x k xs sh sh in

i

mi

, ,max ,max

!( )= = ⋅ =+

=∑ 1

11 , (8)

one obtains the capacity of the subject shared lane

L k q k qsh sh ii

m

= ⋅ = ⋅=∑

1

(9)

Ning Wu 8

Accordingly , the real saturation degree of the shared lane becomes

xqL ksh real

sh

sh, = =

1 (10)

Thereby k is determined implicitly by eq.(8). For n1 = n2 =. .. = ni =. .. = nm = n, i.e., all sub-streams

have the same length of queue space, one gets,

kx

all n n

in

i

m

n

i| =

+

=

+

=

∑

1

1

1

1

(11)

and

Lq

xsh all n n

ii

m

in

i

m

n

i| =

=

+

=

+

=∑

∑1

1

1

1

(12)

For ni with general values the eq.(8) cannot be solved explicitly for k. The solution for k can be found

however according to the Newton-Method iteractively and numerically. The procedure of the

iterations is

k kf kf kj j

j

j+ = −

′1

( )( )

(j = 0, 1, 2,...; k0=1)

with (13)

f k k xin

i

mi( ) ( )= ⋅ −+

=∑ 1

11

This corresponds to

[ ]k k

k x

n k x xj j

j in

i

m

i j in

ii

m

i

i

+

+

=

=

= −⋅ −

+ ⋅ ⋅ ⋅

∑

∑1

1

1

1

1

1

( )

( ) ( ) (j = 0, 1, 2,...; k0=1) (14)

The iterations are convergent for all k > 0.

Ning Wu 9

.

.nm1,i2

.

.ni1,i2

n2,i2

n1,i2

Bi2.......

nsh1,m2

.

.nsh1,i2

nsh1,2

nsh1,1

A

qsh2, Lsh2, xsh2 qsh1,2, Lsh1,2, xsh1,2

Bm2

B2

B1

qsh1,m2, Lsh1,m2, xsh1,m2

q1,i2, L1,i2, x1,i2

qi,i21, Li,i2, xi,i2

q2,i2, L2,i2, x2,i2

qm1,i2, Lm1,i2, xm1,i2

qsh1,i2, Lsh1,i2, xsh1,i2

qsh1,1, Lsh1,1, xsh1,1

Fig.3 - Relationship between shared lanes and their sub- and sub-sub streams

If a sub-stream again consists of several sub-sub-streams, this sub-stream must be considered as a

shared stream itself. One gets accordingly, in analog to the eq.(1), for the merge point A:

P Ps sh s ii

m

, ,2 22 1

2

==∑ (15)

And for the sub merge points Bi2, one obtains:

P Ps sh i s i ii

m i

, , , ,1 2 1 21 1

1 2

==∑ (16)

If one considers the queuing systems in all sub-and sub-sub-streams as M/M/1-queuing systems

respectively, then one gets for the sub-sub-stream with index i1, i2

P xs i i i ini i

, , ,,

1 2 1 211 2= + , (17)

for the sub merge point with the index Bi2 (section between point A and Bi2)

P x P xs sh i sh i s i ii

m

i in

i

mii i

i

, , , , , ,,

1 2 1 2 1 21 1

1

1 21

1 1

121 2

2

= = ==

+

=∑ ∑ , (18)

Ning Wu 10

for the sub-stream with the index i2

P x P P xs i sh in

s sh in

s i ii

m n

i in

i

m n

sh i sh ii sh i

i ii sh i

, , , , , , ,, ,

,

,

,

2 1 21

1 21

1 21 1

1 1

1 21

1 1

1 1

1 2 1 22 1 2

1 22 1 2

= = =⎛⎝⎜

⎞⎠⎟ =

⎛⎝⎜

⎞⎠⎟+ +

=

+

+

=

+

∑ ∑ , (19)

and for the merge point A

P x P P xs sh sh s ii

m

s i ii

m n

i

m

i in

i

m n

i

mi sh i

i ii sh i

, , , , ,

,

,

,

2 2 22 1

2

1 21 1

1 1

2 1

2

1 21

1 1

1 1

2 1

22 1 2

1 22 1 2

= = =⎛⎝⎜

⎞⎠⎟ =

⎛⎝⎜

⎞⎠⎟

= =

+

=

+

=

+

=∑ ∑∑ ∑∑ . (20)

Multiplying the saturation degree of all sub-sub-streams by a factor k and postulating

( )P x k xs sh sh i i

n

i

m n

i

mi i

i sh i

, ,max ,max ,

!,

,

2 2 1 2

1

1 1

1 1

2 1

21 2

2 1 2

1= = ⋅⎡

⎣⎢

⎤

⎦⎥ =

+

=

+

=∑∑ , (21)

the capacity of the total shared stream becomes

L L k q k qsh sh sh i ii

m

i

m i

= = ⋅ = ⋅==∑∑2 2 1 21 1

1

2 1

2 2

, (22)

The eqs.(20), (21), and (22) are the generalized forms of eqs.(7), (8) and (9). Setting m1i2 or m2 equal

to 1, one obtains here the eqs.(7), (8) and (9) again.

For ni1,i2 and nsh1,i2 with general values the iteration procedure for solving k becomes

k kf kf kj j

j

j+ = −

′1

( )( )

(j = 0, 1, 2,...; k0=1)

with (23)

( )f k k xi i

n

i

m n

i

mi i

i sh i

( ) ,,

,

= ⋅⎡

⎣⎢

⎤

⎦⎥ −

+

=

+

=∑∑ 1 2

1

1 1

1 1

2 1

21 2

2 1 2

1

Ning Wu 11

This corresponds to

( )

( ) ( ) ( ) ( )k k

k x

n k x n k x xj j

j i i

n

i

m n

i

m

sh i j i i

n

i

m n

i i j i i

n

i ii

m

i ii sh i

i ii sh i

i ii

+

+

=

+

=

+

= =

= −⋅

⎡

⎣⎢

⎤

⎦⎥ −

+ ⋅ ⋅⎡

⎣⎢

⎤

⎦⎥ ⋅ + ⋅ ⋅ ⋅⎡

⎣⎢⎤⎦⎥

⎧⎨

∑∑

∑ ∑1

1 2

1

1 1

1 1

2 1

2

1 2 1 2

1

1 1

1

1 2 1 2 1 21 1

1

1 22 1 2

1 22 1 2

1 22

1

1 1

,

, , , , ,

,,

,,

,⎪

⎩⎪

⎫⎬⎪

⎭⎪=∑i

m

2 1

2

Analogously we can also treat systems with arbitrarily many levels of sub-sub-streams.

3. PRACTICAL APPLICATIONS OF THE THEORY

3.1. Type 1 of lane combinations (cf. Fig.1b and 4)

The right and left turn streams divide at a point from the crossing stream

nR

nG

nL

A

qsh, Lsh, xshqL, LL, xL

qR LR, xR

qG, LG, xG

Stop line

Fig.4 - Parameters for Type 1 of short traffic lanes

Setting in eqs.(20), (21), and (22) m2 = 1 and i1 = L, G, R, one obtains for Type 1 of short lanes at

intersections without traffic signals the equations for estimating the capacity of the shared lane:

P P P N n x x x xs sh type s i i in

Ln

Gn

Rni L G R

, ,| ( )11 1 1 1= = > = = + +∑ ∑ ∑ + + + + (25)

P x k x k x k x k xs sh type sh in

Ln

Gn

Rni L G R

, ,max ,max

!| ( ) ( ) ( ) ( )1

1 1 1 1 1= = ⋅ = ⋅ + ⋅ + ⋅ =+ + + +∑ (26)

L k q k q q qsh type sh L G R| ( )1 = ⋅ = ⋅ + + (27)

Ning Wu 12

In general, the three streams (L, G, and R) must stop and wait at the same stop line. This means, that

the numbers of the available queue places are equal for all three streams. Setting in this case nL = nG =

nR = n, one obtains

kx x x x

type

inn

Ln

Gn

Rnn

| 1 11 1 1 11

1 1= =

+ +++ + + ++∑ (28)

and

L k q q qq q q

x x xsh type type L G R

L G R

Ln

Gn

Rnn

| | ( )1 1 1 1 11= ⋅ + + =

+ +

+ ++ + ++ (29)

At n = 0 one gets

Lq q qx x xsh type n

L G R

L G R

| ,1 0= =+ ++ +

(30)

That is exactly the well-known shared lane formula from Harders4).

For nL, nG, and nR with general values the iteration procedure for solving k (cf. eq.(14)) yields

k kk x k x k x

n k x x n k x x n k x xj type jj L

nj G

nj R

n

L j Ln

L G j Gn

G R j Rn

R

L G R

L G R+

+ + +

= −⋅ + ⋅ + ⋅ −

+ ⋅ ⋅ ⋅ + + ⋅ ⋅ ⋅ + + ⋅ ⋅ ⋅1 1

1 1 1 11 1 1

|( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

(j = 0, 1, 2,...; k0=1) (31)

With this equation, the Newton-Iteration-Procedure is to be used for determining the subject k. The

capacity of the whole approach can then be obtained according to eq.(9).

Ning Wu 13

3.2. Type 2 of lane combinations (cf. Fig.1b and 5):

Right turn stream merge in the crossing stream before the left turn stream

nR

nG

nL

nGR

A B

qsh, Lsh, xshqL, LL, xL, nLqL, LL, xL

qR, LR, xR

qG, LG, xG

Stop line

Fig. 5 - Parameters for Type 2 of short traffic lanes

Setting in eqs.(20), (21), and (22) i2 = L, GR and i1 = G, R, one obtains for Type 2 of short lanes at

intersections without traffic signals for estimation the capacity of the shared lane:

P x x xs sh type Ln

Gn

Rn nL G R GR

, | ( )21 1 1 1= + ++ + + + (32)

[ ]P x k x k x k xs sh type sh Ln

Gn

Rn n

L G R GR

, ,max ,max

!| ( ) ( ) ( )2

1 1 1 11= = ⋅ + ⋅ + ⋅ =+ + + +

(33)

L k q k q q qsh type sh L G R| ( )2 = ⋅ = ⋅ + + (34)

Also here the three streams must generally stop and wait at the same stop line. The following

relationships exist between the available queue places:

n n nG R G R= = ,

n n nL GR G R= + ,

where nG,R is the common number of queue spaces for left turn and crossing streams.

According to these relationships one gets

P x x xs sh type n n n n n Ln

Gn

Rn n

G R L GR G R

L G R G R GR, , ,| ( )

,

, ,2

1 1 1 1= = +

+ + + += + + (35)

Ning Wu 14

[ ]

( )

P x

k x k x k x

k x x x k

k x x

s sh type n n n n n sh

Ln

Gn

Rn n

Ln

Gn

Rn

n n n n

Ln

Gn

G R L GR G R

L G R G R GR

L G R G R

nG R

GR G R GR G R

L G

, ,max , , ,max|

( ) ( ) ( )

( )

( )

,

, ,

, ,,

, ,

,

2

1 1 1 1

1 1 1

1

1

11

= = +

+ + + +

+ + +

⋅ + + +

+

=

= ⋅ + ⋅ + ⋅

= ⋅ + + ⋅⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

= ⋅ +

+

( )R G R

nG R

GR G R L

x kRn

n n n

+ +

⋅ + +

+ ⋅⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

=

+1 1

111

1

,,

,

!

(36)

Setting

( )

x xn n

x x x

n n n n nn n n

I L

I L

II Gn

Rn

II GR G R GR G R

GR G R L

G R G R

nG R

==

= +

= ⋅ + +

= ⋅ +

+ ++

, ,,

, ,

,

1 1

11

(37)

one obtains for the postulate (eq.(36))

P x k x k xs sh type n n n n n sh In

IIn

G R L GR G R

I II, ,max , , ,max

!| ( ) ( )

,21 1 1= = ++ += = ⋅ + ⋅ = (38)

This means: under the marginal condition that all three sub-streams stop and wait at the same stop line,

the shared lane-system with three sub-streams can be simplified in a shared lane-system with only two

sub-streams.

Furthermore, the capacity under this condition is

L k q q qq q q

xsh type n n n n n L G RL G R

sh realG R L GR G R

| ( )( )

, ,,

,2 = = + = ⋅ + + =+ +

(39)

Ning Wu 15

For nL, nG, and nR with general values the iteration procedure for estimating k yields

k ka

b c dj j+ = −+ ⋅1 , (j = 0,1,2,...; k0=1) (40)

with

[ ]a k x k x k xj Ln

j Gn

j Rn n

L G RGR

= ⋅ + ⋅ + ⋅ −+ + + +( ) ( ) ( )1 1 1 1

1

b n k x xL jn

Ln

LL L= + ⋅ ⋅ ⋅( ) ( )1

[ ]c n k x k xGR j Gn

j Rn n

G RGR

= + ⋅ ⋅ + ⋅+ +( ) ( ) ( )1 1 1

[ ]d n k x x n k x xG j Gn

G R j Rn

RG R= + ⋅ ⋅ ⋅ + + ⋅ ⋅ ⋅( ) ( ) ( ) ( )1 1

3.3. Type 3 of lane combinations (cf. Fig.1b and 6):

Left turn stream merges before the right turn stream in the crossing stream

nR

nG

nL

nLG

A B

qsh, Lsh, xshqL, LL, xL

qR LR, xR

qG, LG, xG

Stop line

Fig. 6 - Parameters for Type 3 of short traffic lanes

Type 3 of short lanes is symmetrically to Type 2. Analogously to eqs.(32), (33), and (34) one obtains

here

P x x xs sh type Ln

Gn n

RnL G LG R

, | ( )31 1 1 1= + ++ + + + (41)

[ ]P x k x k x k xs sh type sh Ln

Gn n

RnL G LG R

, ,max ,max

!| ( ) ( ) ( )3

1 1 1 1 1= = ⋅ + ⋅ + ⋅ =+ + + + (42)

L k q k q q qsh type sh L G R| ( )3 = ⋅ = ⋅ + + (43)

Ning Wu 16

Under the conditions (nL,G is the common number of queue places for crossing and left turn streams)

n n nL G L G= = ,

n n nR LG L G= + ,

one obtains

P x x xs sh type n n n n n Ln

Gn n

Rn

L G R LG L G

L G L G LG R, , ,| ( )

,

, ,3

1 1 1 1= = +

+ + + += + + (44)

and

P x k x k xs sh type n n n n n sh In

IIn

G R L GR G R

I II, ,max , , ,max

!| ( ) ( )

,31 1 1= = ++ += = ⋅ + ⋅ = (45)

with

( )x x x

n n n n nn n n

x xn n

I Ln

Gn

I LG L G LG L G

LG L G R

II R

II R

L G L G

nL G

= +

= ⋅ + +

= ⋅ +

==

+ ++

, ,,

, ,

,

1 1

11

(46)

For nL, nG, and nR with general values the procedure of the iterations for estimating k yields

k ka

b c dj j+ = −⋅ +1 , (j = 0, 1, 2,...; k0=1) (47)

with

[ ]a k x k x k xj Ln

j Gn n

j RnL G

LGR= ⋅ + ⋅ + ⋅ −+ + + +( ) ( ) ( )1 1 1 1 1

[ ]b n k x k xLG j Ln

j Gn n

L GLG

= + ⋅ ⋅ + ⋅+ +( ) ( ) ( )1 1 1

[ ]c n k x x n k x xL j Ln

L G j Gn

GL G= + ⋅ ⋅ ⋅ + + ⋅ ⋅ ⋅( ) ( ) ( ) ( )1 1

d n k x xR j Rn

RR= + ⋅ ⋅( ) ( )1

Ning Wu 17

3.4. Type 4 of lane combinations (cf. Fig.1a and 7)

Left turn stream on the major street

nR,H=0

nG,H=0

nL≠0

nGR

A

qsh, Lsh, xshqL, LL, xL, nLqL,H, LL,H, xL,H

qR,H, LR,H, xR,H

qG,H, LG,H, xG,H

Fig. 7 - Parameters for Type 4 of short lanes on major streets

Setting in the eqs.(20), (21), and (22) m2 = 1 and i1 = L,H, G,H, R,H, with nG,H =0 and nR,H = 0, one

obtains here:

P P P N n x

x x x

x x x

s sh type s i i in

L Hn

G Hn

R Hn

L Hn

G H R H

i

L H G H R H

L H

, ,

, , ,

, , ,

| ( ), , ,

,

41

1 1 1

1

= = > =

= + +

= + +

∑ ∑ ∑ +

+ + +

+

(48)

P x k x k x k x

k x k x xs sh type sh L H

nG H R H

L Hn

G H R H

L H

L H

, ,max ,max , , ,

, , ,

!

| ( )

( ) ( )

,

,

41

1

1

= = ⋅ + ⋅ + ⋅

= ⋅ + ⋅ +

=

+

+ (49)

L k q k q q qsh type sh L G R| ( )4 = ⋅ = ⋅ + + (50)

Setting

x xn nx x xn

I L H

I L H

II G H R H

II

=

=

= +

=

,

,

, ,

0

(51)

one obtains again for the postulate (eq.(49))

P x k x k xs sh type sh In

III

, ,max ,max

!| ( )4

1 1= = ⋅ + ⋅ =+ (52)

Ning Wu 18

For nL,H with general values the iteration procedure for estimating k becomes

k kk x k x k x

n k x x x xj type jj L H

nj G H j R H

L j L Hn

L H G H R H

L H

L H+

+

= −⋅ + ⋅ + ⋅ −

+ ⋅ ⋅ ⋅ + +1 4

1 11

|( )

( ) ( ), , ,

, , , ,

,

,

(j = 0, 1, 2,...; k0=1) (53)

For the system with two sub-streams, a graph (see appendix) for estimating the value of xsh,real = 1 / k is

prepared. With this graph the iterations for estimating xsh,real =1 / k can be done without computers.

The input data of the graph are the saturation degrees of the sub-streams xI, xII, and the numbers of

queue places for the sub-streams nI, nII.

The usage of this graph is explained using an example.

Example:

Estimate of the capacity of a shared lane on the major street (cf. Fig. 7)

Given:

Capacity of the left turn stream: LL,H = 500 [veh/h]

Capacity of the through traffic stream: LG,H = 1800 [veh/h]

Capacity of the right turn stream : LR,H = 1600 [veh/h]

Traffic flow of the left turn stream: qL,H = 500 [veh/h]

Traffic flow of the through traffic stream: qG,H = 450 [veh/h]

Traffic flow of the right turn stream: qR,H = 80 [veh/h]

Number of queue places for the left turn stream: nL,H = 2 [veh]

Number of queue places for the through traffic stream: nG,H 0 [veh]

Number of queue places for the right turn stream: nR,H = 0 [veh]

Ning Wu 19

Solution:

According to the eq.(51) one obtains xI = xG,H + xR,H

= 450 / 1800 +80 / 1600

= 0.25 + 0.05

= 0.3

xII = xL,H

= 250 / 500

= 0.5

nI = xG,H = xR,H

= 0

nII = xL,H

= 2

(Note, that the order of I and II is unimportant here)

Estimating xsh,real from the graph (see appendix):

Step 1: Draw two vertical lines through xI = 0.3 and xII = 0.5

Step 2: Go along the line of the smaller x (here xI = 0.3, important for the convergence) upwards to

the line of the initial value of xsh,real|loop 0 of the iterations: Point A (here with xsh,real|loop 0 = 1)

Step 3: Go horizontally to the right until to the line with the value nI: Point 1 (here with nI = 0)

Step 4: Go vertically upwards to the line with the value nII: Point 2 (here with nII = 2)

Step 5: Go horizontally again to the left to the line with the value xII: Point 3 (here with xII = 0.5).

Step 6: Draw a line through point 3 and the origin (corresponding to the value of xsh,real|loop 1) and

go along this line downwards to the line with the value xI: Point 4 (here with xI = 0.3)

Step 7: Repeat the steps 3 through 7 until the line drawn in the step 6 can no longer be

distinguished from the preceding loop or if the precision expected is reached

Step 8: Read the final value on the axis of xsh,real: Point B (here with xsh,real = 0.625)

Thus, the capacity of the shared lane yields

Ning Wu 20

Lq

xq q q

x

shsh

sh real

L G R

sh real

=

=+ +

=+ +

=

,

,

.[

250 450 800 625

1248 veh / h]

so that the shared lane has a total capacity of 1248 [veh/h].

3.5. Flared lane at minor approaches

nF,rightL+G

R

Fig.8 - Right flared approach

3.5.1. Right flared lane at minor approaches

A special application of eqs.(33) and (42) is the so-called flared lanes (cf. Fig.8). For the right flared

approach (right turn stream passes by the left + crossing stream) the following relationships are valid:

nL = nG = 0

nLG = nR = nF,right

Accordingly, one gets the postulate

[ ]P x k x k x k xs sh F right sh F right L F right G

n

F right RnF right F right

, , .max , , ,

!| ( ) ( ) ( ), ,= = ⋅ + ⋅ + ⋅ =

+ +1 1 1

Solving this equation for kF,right, one obtains

Ning Wu 21

kx x x

F right

L Gn

Rnn F right F rightF right

,( ) , ,,

=+ ++ ++

11 11

(54)

and

L k qq q q

x x xF right F right i

L G R

L Gn

Rnn F right F rightF right

, ,( ) , ,,

= ⋅ =+ +

+ +∑ + ++ 1 11

(55)

For nF,right = 1 it becomes

L q q q

x x x

q q q

x x xF rechts n

L G R

L G R

L G R

L G RF rechts, |

( ) ( ), = + ++=

+ +

+ +=

+ +

+ +1 1 1 1 11 1 2 22

(56)

nF,leftG+R

L

Fig.9 - Left flared approach

3.5.2. Left flared lane at minor approaches

Analogously, one obtains for the left flared approach (left turn stream passes by the right + crossing

stream, Fig.9)

Lq q q

x x xF left

L G R

Ln

G Rnn F left F leftF left

,, ,, ( )

=+ +

+ ++ ++ 1 11 (57)

and for nF,left = 1

Lq q q

x x xF left n

L G R

L G RF left, |

( ), = =+ +

+ +1 2 22

(58)

Ning Wu 22

For nF = 0 the eqs.(55) and (57) yield

L Lq q q

x x x

q q q

x x x

q q qx x xF right n F left n

L G R

L G R

L G R

L G R

L G R

L G RF right F left, ,| |

( ) ( ), ,= = + ++ + ++= =

+ +

+ +=

+ +

+ +=

+ ++ +0 0 0 1 0 10 1 0 1 0 10 1

One gets again the shared lane formula from Harders4).

3.5.3. Mixed flared lane at minor approaches

Figs.8 and 9 show the two possibilities, how a flared approach can be used by vehicles. However, it is

not easy to forecast, how the vehicles in reality would use the flared approach. Here, only the driver

behavior of the crossing vehicles is decisive for the calculation, because the right and left turn vehicles

always pass by each other at a flared approach. The decision of a crossing driver, whether he passes

by a waiting left turn vehicle or by a waiting right turn vehicle, determines the configuration of the

flared approach. If a crossing driver passes on the left of a waiting right turn vehicle, the approach is a

right flared approach (because the left turn vehicles must pass by the waiting right turn vehicle also).

If a crossing driver passes on the right of a waiting left turn vehicle, the approach is a left flared

approach (because the right turn vehicles must pass by the waiting left turn vehicle also). If a crossing

driver arrives while another crossing vehicle is waiting alone on the stop line, the approach could also

be considered as a right flared approach (because in this case only the right turn vehicles may drive by

on the right. As an approximation one can assume, that the probabilities, whether the approach is used

as a left or right flared approach, are proportional to the corresponding saturation degrees of the traffic

streams. According to this consideration an equation for estimating the capacity of the flared approach

with mixed configuration, which treats the approach both as a left flared approach and a right flared

approach, can be represented by

L Lx

x x xL

x xx x xF mix F left

L

L G RF right

G R

L G R, , ,= ⋅

+ ++ ⋅

++ +

(59)

according to the saturation degrees, respectively.

Inserting eqs.(55) and (57) in the eq.(59) and setting nF,left = nF,right = nF, one obtains

Ning Wu 23

Lq q q

x x x

xx x x

q q q

x x x

x xx x x

x

x x x

x x

x x x

F mixL G R

Ln

G Rnn

L

L G R

L G R

L Gn

Rnn

G R

L G R

L

Ln

G Rnn

G R

L Gn

Rnn

F FF F FF

F FF F FF

,( ) ( )

( ) ( )

=+ +

+ +⋅

+ ++

+ +

+ +⋅

++ +

=+ +

++

+ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟ ⋅

+ ++ + ++

+ ++ + ++

1 11 1 11

1 11 1 11

q q qx x x

x

x x x

x x

x x xL

L G R

L G R

L

Ln

G Rnn

G R

L Gn

Rnn n

F FF F FF

+ ++ +

=+ +

++

+ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟ ⋅+ ++ + ++ =1 11 1 11 0

( ) ( )

(60)

where

Lq q qx x xn

L G R

L G R= =

+ ++ +0

is the capacity of the shared lane for the case n = 0 (corresponding to Harders4) formula).

For nF = 1, the eq.(60) yields

Lx

x x x

x x

x x x

q q qx x x

x

x x x

x x

x x xL

F mix nL

L G R

G R

L G R

L G R

L G R

L

L G R

G R

L G R

n

F, |( ) ( )

( ) ( )

=

=

=+ +

++

+ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟ ⋅

+ ++ +

=+ +

++

+ +

⎛

⎝⎜⎜

⎞

⎠⎟⎟ ⋅

1 2 22 2 22

2 22 2 220

(61)

In Fig.11 a comparison of increases of capacity caused by the flaring of the approach is presented. For

this comparison the north bound approach of a standard intersection is calculated. The capacities of

the separate traffic streams are obtained according to the German Highway Capacity Manual2). The

traffic flow of this intersection is shown in Fig.10. The calculation yields the parameters xL = 0.33, xG

= 0.46, and xR = 0.05 for the subject approach. These parameters characterized qualitatively most of

the real traffic conditions at approaches of intersections without traffic signals.

Ning Wu 24

100300500

400

300

10030050

400

50 150 50

50 150 50

300

Fig.10 - Traffic flow of the test example

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

0 1 2 3 4

length of flared area n [veh]

Fact

or o

f cap

acity

incr

ease

k [-

]

right flaring

left flaring

mixed flaring

Fig. 11 - Increases of capacity caused by flaring

xL=0.33

xG=0.46

xR=0.05

Ning Wu 25

One can recognize clearly, that the left flaring causes a manifold increase of capacity compared to the

right flaring. For the left flaring, the example has at nF,left = 1 a capacity increase of 38% compared to

that with nF,left = 0. For the right flaring it is barely 6%. The mixed using of the flared area, which is

more realistic than the pure left and/or right flaring, delivers approximately 18% increase of capacity.

The value of the mixed flaring corresponds very well to the measurements in the technical report from

Kyte et al8) (see there, Fig.8.7).

4. EXAMINATION OF THE THEORY BY SIMULATION

To check the derived theory, different combinations of shared-short lanes are simulated in the style of

KNOSIMO7). Altogether 95 traffic flow and lane variations were simulated. The simulation results

are presented in Fig.12, together with the theoretical values. The key statistical values of this

comparison are assembled in Table 1. It shows that the relationships between both parameters are

narrowly correlated. Accordingly, one can be certain for the correctness of the derived theory.

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

Simulated capacity [veh/h]

Cal

cula

ted

capa

city

[veh

/h]

Fig. 12 - comparison of calculated and simulated capacities

Ning Wu 26

Multiple correlation coefficient [-] 0.993

Certainty [-] 0.985

Adjusted certainty [-] 0.985

Standard errors [veh/h] 24.42

Observations [-] 95

Tab.1 - Key statistical values of the comparison

5. SUMMARY AND OUTLOOK

The theory derived here delivers a general approach for estimating the capacity of shared-short lanes

at intersections without traffic signals. This theory considers the length of short lanes and fills out a

gap in the current calculation procedures for intersections without traffic signals.

The derivation of this theory presumes, that the queuing systems at intersection without traffic signals

can be approximately considered as M/M/1-queuing systems. This could lead to a minor deviation of

the results from reality. The simulation results show however, that this deviation is negligibly small

and statistical not significant.

For practical applications the eqs.(55), (57), and (60) are most important. With these three equations

the capacity of minor approaches at intersection without traffic signals with left, right and mixed

flaring can be on the simplest way and exactly determined.

It shows, that most minor approaches with three traffic streams (and lanes) can be mathematically

simplified in a system with two sub-streams. For this system with two sub-streams a graph is

prepared, which makes the manual calculation of the capacity possible also under complicated lane

combinations.

For shared-short lanes with arbitrary lane combinations a general implicit equation for estimating the

capacity is given (eqs.(21) and (22)). For the solution of the implicit eq.(21) the Newton-Iteration-

Method can be used (eq.(23)). The eq.(24) describes the concrete procedure of the iterations. With

this procedure all possible shared-short lane combinations at intersections without traffic signals can

Ning Wu 27

be determined without large expenses using computers. If a worksheet program is available, one can

also use the so-called SOLVER (by Excel) to solve the eq.(21).

As a summary, all possible configurations of shared-short lanes and their solutions are assembled in

Table 2.

In this paper, it is presupposed for estimating the capacity of shared lanes, that the traffic flows of all

sub-streams increase proportionally to their original traffic flows. All sub-streams were multiplied by

the same factor k. It is also possible however, to determine the capacity of a certain sub-stream by

using fixed traffic flows for all other sub-streams.

The theory should be expanded to intersections with traffic signals

Ning Wu 28

Tab.2 - Capacity of shared - short lanes Lsh Case Figure Equation for k1) Notation

1

.

.nm

.

.ni

n2

n1

A

qsh, Lsh, xshq1, L1, x1

qi, Li, xi

q2, L2, x2

qm, Lm, xm

eqs.(13) and (14), iteration procedure. For all ni=n, eq.(11), explicit2).

Generalized case with one level of sub-streams

2

.

.nm1,i2

.

.ni1,i2

n2,i2

n1,i2

Bi2.......

nsh1,m2

.

.nsh1,i2

nsh1,2

nsh1,1

A

qsh2, Lsh2, xsh2 qsh1,2, Lsh1,2, xsh1,2

Bm2

B2

B1

qsh1,m2, Lsh1,m2, xsh1,m2

q1,i2, L1,i2, x1,i2

qi,i21, Li,i2, xi,i2

q2,i2, L2,i2, x2,i2

qm1,i2, Lm1,i2, xm1,i2

qsh1,i2, Lsh1,i2, xsh1,i2

qsh1,1, Lsh1,1, xsh1,1

eqs.(23) and (24) , iteration procedure.

Generalized case with two levels of sub-streams

3

nR

nG

nL

A

qsh, Lsh, xshqL, LL, xL

qR LR, xR

qG, LG, xG

Stop line

eq.(31) , iteration procedure. For ni=n, eq.(28), explicit2).

4

nR

nG

nL

nGR

A B

qsh, Lsh, xshqL, LL, xL, nLqL, LL, xL

qR, LR, xR

qG, LG, xG

Stop line

eq.(40) , iteration procedure.

Graph applicable

5

nR

nG

nL

nLG

A B

qsh, Lsh, xshqL, LL, xL

qR LR, xR

qG, LG, xG

Stop line

eq.(47) , iteration procedure.

Graph applicable

6

nR,H=0

nG,H=0

nL≠0

nGR

A

qsh, Lsh, xshqL, LL, xL, nLqL,H, LL,H, xL,H

qR,H, LR,H, xR,H

qG,H, LG,H, xG,H

eq.(53) , iteration procedure.

Graph applicable

7 nF,right

L+G

R

eq.(54), explicit2). Right flared lane

8 nF,left

G+R

L

eq.(57), explicit2). Left flared lane

9 Case 7 + Case 8 eq.(60), explicit2). Mixed flared lane 1) L k q k qsh sh i= ⋅ = ⋅∑ , 2)Lsh is directly available

Ning Wu 29

REFERENCES

1) HCM. Highway Capacity Manual. Transportation Research Board, Special Report 209, new

edition, Washington 1994.

2) D-HCM (German Highway Capacity Manual). Verfahren für die Berechnung der

Leistungsfähigkeit und Qualität des Verkehrsablaufes auf Straßen (Deutsches HCM).

Schriftenreihe "Forschung Straßenbau und Straßenverkehrstechnik", Heft 669, 1964.

3) Worksheet for calculating capacities at unsignalised intersections. Merkblatt zur Berechnung

der Leistungsfähigkeit von Knotenpunkten ohne Lichtsignalanlagen. Herausgeber:

Forschungsgesellschaft für Straßen- und Verkehrswesen. 1991.

4) J. Harders. Die Leistungsfähigkeit nicht signalgeregelter städtischer Verkehrsknoten.

Schriftenreihe "Straßenbau und Straßenverkehrstechnik", Heft 76, 1968.

5) W. Siegloch. Die Leistungsermittlung an Knotenpunkten ohne Lichtsignalsteuerung.

Schriftenreihe "Straßenbau und Straßenverkehrstechnik", Heft 154, 1973.

6) N. Wu. An Approximation for the Distribution of Queue Lengths at Unsignalised Intersections.

Proceeding of the second International Symposium on Highway Capacity, Sydney, 1994. In

Akcelik, R. (ed.), volume 2, pp. 717-736.

7) M. Grossmann. Methoden zur Berechnung und Beurteilung von Leistungsfähigkeit und

Verkehrsqualität an Knotenpunkten ohne Lichtsignalanlagen. Ruhr-Universität Bochum,

Schriftenreihe Lehrstuhl für Verkehrswesen, Heft 9, 1991.

8) Kyte, M. et al. Capacity and level of service at unsignalised intersections. Final report, volume 2

- Two-way stop-controlled intersections. National cooperative highway research program, Project

3-46. Dec. 1995.

Ning Wu

30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Saturation degree of the sub - streams xI, xII [-]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Saturation degree of the shared lane xsh,real [-]

1

23

4

5 67 8

9

1

23

45

6789

xI = 0.3 xII = 0.5

nI =0

nII =2xsh,real=0.625

xsh,real

xI, nII

xI, nI

nII = 0iterations

A

B

nI = 0

1

23

4

Queue places

Appendix - Capacity of shared - short lanes with two sub - streams : Iterations for xx

xx

I

sh real

n

II

sh real

nI II

, ,

⎛

⎝⎜

⎞

⎠⎟ +

⎛

⎝⎜

⎞

⎠⎟ =

+ +1 1

1

Example : xI = 0.3, xII = 0.5, nI = 0, nII = 2 ⇒ Iterations from point A through 1,2,3,4, ... to B ⇒ xsh,real = 0.625 (cf. page 18)

Ning Wu Listing of Tables and Figures

Tab. 1 - Key statistical values of the comparison

Tab.2 - Capacity of shared - short lanes Lsh

Fig.1a - Possible queues at the approaches of unsignalised intersections

Fig.1b - Combination forms of short traffic lanes

Fig.2 - Relationship between a shared lane and its sub - streams

Fig.3 - Relationship between shared lanes and their sub - and sub - sub streams

Fig.4 - Parameters for the type 1 of short traffic lanes

Fig. 5 - Parameters for the type 2 of short traffic lanes

Fig. 6 - Parameters for the type 3 of short traffic lanes

Fig. 7 - Parameters for the type 4 of short lanes on major streets

Fig.8 - Right flared approach

Fig.9 - Left flared approach

Fig.10 - Traffic flow of the test example

Fig. 11 - Increases of capacity caused by flaring

Fig. 12 - comparison of calculated and simulated capacities

Appendix - Capacity of shared - short lanes with two sub - streams : Iterations